Proposition 2.24 (See e.g. Example 2.6 in [1]}) 
Let \((L, [\cdot ,\cdot ]_{L}, \rho _L)\)  be a Lie algebroid, and consider the vector bundle \(L\oplus L^{*}\) .
Denote sections of \(L\oplus L^{*}\)  by \((X,\alpha ), (Y,\beta )\)  etc., and introduce
the following maps:
\(& \pi _{L\oplus L^{*}}((X,\alpha ))=X , \\& \langle (X,\alpha ),(Y,\beta )\rangle _{L\oplus L^{*}} = \alpha (Y)+\beta (X),  \\& \llbracket (X,\alpha ), (Y,\beta ) \rrbracket _{L\oplus L^{*}} = ([X,Y]_L, £^L_X\beta -£^L_Y\alpha +{\rm d}^L(i_{Y}\alpha )) \) 
